:: On the {H}ausdorff Distance Between Compact Subsets
:: by Adam Grabowski
::
:: Received January 27, 2003
:: Copyright (c) 2003-2021 Association of Mizar Users

registration

let M be non empty MetrSpace;

cluster TopSpaceMetr M -> T_2 ;

coherence
TopSpaceMetr M is T_2 by PCOMPS_1:34;

end;

theorem Th1: :: HAUSDORF:1

for x, y being Real st x >= 0 & max (x,y) = 0 holds
x = 0

proof end;

theorem Th2: :: HAUSDORF:2

for M being non empty MetrSpace
for x being Point of M holds (dist x) . x = 0

proof end;

theorem Th3: :: HAUSDORF:3

for M being non empty MetrSpace
for P being Subset of (TopSpaceMetr M)
for x being Point of M st x in P holds
0 in (dist x) .: P

proof end;

theorem Th4: :: HAUSDORF:4

for M being non empty MetrSpace
for P being Subset of (TopSpaceMetr M)
for x being Point of M
for y being Real st y in (dist x) .: P holds
y >= 0

proof end;

theorem Th5: :: HAUSDORF:5

for M being non empty MetrSpace
for P being Subset of (TopSpaceMetr M)
for x being set st x in P holds
(dist_min P) . x = 0

proof end;

theorem Th6: :: HAUSDORF:6

for M being non empty MetrSpace
for p being Point of M
for A being Subset of (TopSpaceMetr M) holds
( p in Cl A iff for r being Real st r > 0 holds
ex q being Point of M st
( q in A & dist (p,q) < r ) )

proof end;

theorem Th7: :: HAUSDORF:7

for M being non empty MetrSpace
for P being non empty Subset of (TopSpaceMetr M)
for x being Point of M holds
( (dist_min P) . x = 0 iff for r being Real st r > 0 holds
ex p being Point of M st
( p in P & dist (x,p) < r ) )

proof end;

theorem Th8: :: HAUSDORF:8

for M being non empty MetrSpace
for P being non empty Subset of (TopSpaceMetr M)
for x being Point of M holds
( x in Cl P iff (dist_min P) . x = 0 )

proof end;

theorem Th9: :: HAUSDORF:9

for M being non empty MetrSpace
for P being non empty closed Subset of (TopSpaceMetr M)
for x being Point of M holds
( x in P iff (dist_min P) . x = 0 )

proof end;

theorem Th10: :: HAUSDORF:10

for A being non empty Subset of R^1 ex X being non empty Subset of REAL st
( A = X & lower_bound A = lower_bound X )

proof end;

theorem Th11: :: HAUSDORF:11

for A being non empty Subset of R^1 ex X being non empty Subset of REAL st
( A = X & upper_bound A = upper_bound X )

proof end;

theorem Th12: :: HAUSDORF:12

for M being non empty MetrSpace
for P being non empty Subset of (TopSpaceMetr M)
for x being Point of M
for X being Subset of REAL st X = (dist x) .: P holds
X is bounded_below

proof end;

theorem Th13: :: HAUSDORF:13

for M being non empty MetrSpace
for P being non empty Subset of (TopSpaceMetr M)
for x, y being Point of M st y in P holds
(dist_min P) . x <= dist (x,y)

proof end;

theorem Th14: :: HAUSDORF:14

for M being non empty MetrSpace
for P being non empty Subset of (TopSpaceMetr M)
for r being Real
for x being Point of M st ( for y being Point of M st y in P holds
dist (x,y) >= r ) holds
(dist_min P) . x >= r

proof end;

theorem Th15: :: HAUSDORF:15

for M being non empty MetrSpace
for P being non empty Subset of (TopSpaceMetr M)
for x, y being Point of M holds (dist_min P) . x <= (dist (x,y)) + ((dist_min P) . y)

proof end;

theorem Th16: :: HAUSDORF:16

for M being non empty MetrSpace
for P being Subset of (TopSpaceMetr M)
for Q being non empty Subset of M st P = Q holds
(TopSpaceMetr M) | P = TopSpaceMetr (M | Q)

proof end;

theorem Th17: :: HAUSDORF:17

for M being non empty MetrSpace
for A being Subset of M
for B being non empty Subset of M
for C being Subset of (M | B) st A = C & C is bounded holds
A is bounded

proof end;

theorem :: HAUSDORF:18

for M being non empty MetrSpace
for B being Subset of M
for A being Subset of (TopSpaceMetr M) st A = B & A is compact holds
B is bounded

proof end;

theorem Th19: :: HAUSDORF:19

for M being non empty MetrSpace
for P being non empty Subset of (TopSpaceMetr M)
for z being Point of M ex w being Point of M st
( w in P & (dist_min P) . z <= dist (w,z) )

proof end;

registration

let M be non empty MetrSpace;
let x be Point of M;

cluster dist x -> continuous ;

coherence
dist x is continuous by WEIERSTR:16;

end;

registration

let M be non empty MetrSpace;
let X be non empty compact Subset of (TopSpaceMetr M);

cluster dist_max X -> continuous ;

coherence
dist_max X is continuous by WEIERSTR:24;

cluster dist_min X -> continuous ;

coherence
dist_min X is continuous by WEIERSTR:27;

end;

Lm1: for M being non empty MetrSpace
for P being non empty Subset of (TopSpaceMetr M)
for x being Point of M
for X being Subset of REAL st X = (dist x) .: P & P is compact holds
X is bounded_above

proof end;

theorem Th20: :: HAUSDORF:20

for M being non empty MetrSpace
for P being non empty Subset of (TopSpaceMetr M)
for x, y being Point of M st y in P & P is compact holds
(dist_max P) . x >= dist (x,y)

proof end;

theorem :: HAUSDORF:21

for M being non empty MetrSpace
for P being non empty Subset of (TopSpaceMetr M)
for z being Point of M st P is compact holds
ex w being Point of M st
( w in P & (dist_max P) . z >= dist (w,z) )

proof end;

theorem Th22: :: HAUSDORF:22

for M being non empty MetrSpace
for P, Q being non empty Subset of (TopSpaceMetr M)
for z being Point of M st P is compact & Q is compact & z in Q holds
(dist_min P) . z <= max_dist_max (P,Q)

proof end;

theorem Th23: :: HAUSDORF:23

for M being non empty MetrSpace
for P, Q being non empty Subset of (TopSpaceMetr M)
for z being Point of M st P is compact & Q is compact & z in Q holds
(dist_max P) . z <= max_dist_max (P,Q)

proof end;

theorem :: HAUSDORF:24

for M being non empty MetrSpace
for P, Q being non empty Subset of (TopSpaceMetr M)
for X being Subset of REAL st X = (dist_max P) .: Q & P is compact & Q is compact holds
X is bounded_above

proof end;

theorem Th25: :: HAUSDORF:25

for M being non empty MetrSpace
for P, Q being non empty Subset of (TopSpaceMetr M)
for X being Subset of REAL st X = (dist_min P) .: Q & P is compact & Q is compact holds
X is bounded_above

proof end;

theorem :: HAUSDORF:26

for M being non empty MetrSpace
for P being non empty Subset of (TopSpaceMetr M)
for z being Point of M st P is compact holds
(dist_min P) . z <= (dist_max P) . z

proof end;

theorem Th27: :: HAUSDORF:27

for M being non empty MetrSpace
for P being non empty Subset of (TopSpaceMetr M) holds (dist_min P) .: P = {0}

proof end;

theorem Th28: :: HAUSDORF:28

for M being non empty MetrSpace
for P, Q being non empty Subset of (TopSpaceMetr M) st P is compact & Q is compact holds
max_dist_min (P,Q) >= 0

proof end;

theorem Th29: :: HAUSDORF:29

for M being non empty MetrSpace
for P being non empty Subset of (TopSpaceMetr M) holds max_dist_min (P,P) = 0

proof end;

theorem :: HAUSDORF:30

for M being non empty MetrSpace
for P, Q being non empty Subset of (TopSpaceMetr M) st P is compact & Q is compact holds
min_dist_max (P,Q) >= 0

proof end;

theorem Th31: :: HAUSDORF:31

for M being non empty MetrSpace
for Q, R being non empty Subset of (TopSpaceMetr M)
for y being Point of M st Q is compact & R is compact & y in Q holds
(dist_min R) . y <= max_dist_min (R,Q)

proof end;

definition

let M be non empty MetrSpace;
let P, Q be Subset of (TopSpaceMetr M);

func HausDist (P,Q) -> Real equals :: HAUSDORF:def 1
max ((max_dist_min (P,Q)),(max_dist_min (Q,P)));

coherence
max ((max_dist_min (P,Q)),(max_dist_min (Q,P))) is Real ;
commutativity
for b₁ being Real
for P, Q being Subset of (TopSpaceMetr M) st b₁ = max ((max_dist_min (P,Q)),(max_dist_min (Q,P))) holds
b₁ = max ((max_dist_min (Q,P)),(max_dist_min (P,Q))) ;

end;

:: deftheorem defines HausDist HAUSDORF:def 1 :
for M being non empty MetrSpace
for P, Q being Subset of (TopSpaceMetr M) holds HausDist (P,Q) = max ((max_dist_min (P,Q)),(max_dist_min (Q,P)));

theorem Th32: :: HAUSDORF:32

for M being non empty MetrSpace
for Q, R being non empty Subset of (TopSpaceMetr M)
for y being Point of M st Q is compact & R is compact & y in Q holds
(dist_min R) . y <= HausDist (Q,R)

proof end;

theorem Th33: :: HAUSDORF:33

for M being non empty MetrSpace
for P, Q, R being non empty Subset of (TopSpaceMetr M) st P is compact & Q is compact & R is compact holds
max_dist_min (P,R) <= (HausDist (P,Q)) + (HausDist (Q,R))

proof end;

theorem :: HAUSDORF:34

for M being non empty MetrSpace
for P, Q, R being non empty Subset of (TopSpaceMetr M) st P is compact & Q is compact & R is compact holds
max_dist_min (R,P) <= (HausDist (P,Q)) + (HausDist (Q,R))

proof end;

theorem Th35: :: HAUSDORF:35

for M being non empty MetrSpace
for P, Q being non empty Subset of (TopSpaceMetr M) st P is compact & Q is compact holds
HausDist (P,Q) >= 0

proof end;

theorem :: HAUSDORF:36

for M being non empty MetrSpace
for P being non empty Subset of (TopSpaceMetr M) holds HausDist (P,P) = 0 by Th29;

theorem Th37: :: HAUSDORF:37

for M being non empty MetrSpace
for P, Q being non empty Subset of (TopSpaceMetr M) st P is compact & Q is compact & HausDist (P,Q) = 0 holds
P = Q

proof end;

theorem Th38: :: HAUSDORF:38

for M being non empty MetrSpace
for P, Q, R being non empty Subset of (TopSpaceMetr M) st P is compact & Q is compact & R is compact holds
HausDist (P,R) <= (HausDist (P,Q)) + (HausDist (Q,R))

proof end;

definition

let n be Element of NAT ;
let P, Q be Subset of (TOP-REAL n);

func max_dist_min (P,Q) -> Real means :: HAUSDORF:def 2
ex P9, Q9 being Subset of (TopSpaceMetr (Euclid n)) st
( P = P9 & Q = Q9 & it = max_dist_min (P9,Q9) );

existence
ex b₁ being Real ex P9, Q9 being Subset of (TopSpaceMetr (Euclid n)) st
( P = P9 & Q = Q9 & b₁ = max_dist_min (P9,Q9) )

proof end;

uniqueness
for b₁, b₂ being Real st ex P9, Q9 being Subset of (TopSpaceMetr (Euclid n)) st
( P = P9 & Q = Q9 & b₁ = max_dist_min (P9,Q9) ) & ex P9, Q9 being Subset of (TopSpaceMetr (Euclid n)) st
( P = P9 & Q = Q9 & b₂ = max_dist_min (P9,Q9) ) holds
b₁ = b₂ ;

end;

:: deftheorem defines max_dist_min HAUSDORF:def 2 :
for n being Element of NAT
for P, Q being Subset of (TOP-REAL n)
for b₄ being Real holds
( b₄ = max_dist_min (P,Q) iff ex P9, Q9 being Subset of (TopSpaceMetr (Euclid n)) st
( P = P9 & Q = Q9 & b₄ = max_dist_min (P9,Q9) ) );

definition

let n be Element of NAT ;
let P, Q be Subset of (TOP-REAL n);

func HausDist (P,Q) -> Real means :Def3: :: HAUSDORF:def 3
ex P9, Q9 being Subset of (TopSpaceMetr (Euclid n)) st
( P = P9 & Q = Q9 & it = HausDist (P9,Q9) );

existence
ex b₁ being Real ex P9, Q9 being Subset of (TopSpaceMetr (Euclid n)) st
( P = P9 & Q = Q9 & b₁ = HausDist (P9,Q9) )

proof end;

uniqueness
for b₁, b₂ being Real st ex P9, Q9 being Subset of (TopSpaceMetr (Euclid n)) st
( P = P9 & Q = Q9 & b₁ = HausDist (P9,Q9) ) & ex P9, Q9 being Subset of (TopSpaceMetr (Euclid n)) st
( P = P9 & Q = Q9 & b₂ = HausDist (P9,Q9) ) holds
b₁ = b₂ ;
commutativity
for b₁ being Real
for P, Q being Subset of (TOP-REAL n) st ex P9, Q9 being Subset of (TopSpaceMetr (Euclid n)) st
( P = P9 & Q = Q9 & b₁ = HausDist (P9,Q9) ) holds
ex P9, Q9 being Subset of (TopSpaceMetr (Euclid n)) st
( Q = P9 & P = Q9 & b₁ = HausDist (P9,Q9) ) ;

end;

:: deftheorem Def3 defines HausDist HAUSDORF:def 3 :
for n being Element of NAT
for P, Q being Subset of (TOP-REAL n)
for b₄ being Real holds
( b₄ = HausDist (P,Q) iff ex P9, Q9 being Subset of (TopSpaceMetr (Euclid n)) st
( P = P9 & Q = Q9 & b₄ = HausDist (P9,Q9) ) );

theorem :: HAUSDORF:39

for n being Element of NAT
for P, Q being non empty Subset of (TOP-REAL n) st P is compact & Q is compact holds
HausDist (P,Q) >= 0

proof end;

theorem :: HAUSDORF:40

for n being Element of NAT
for P being non empty Subset of (TOP-REAL n) holds HausDist (P,P) = 0

proof end;

theorem :: HAUSDORF:41

for n being Element of NAT
for P, Q being non empty Subset of (TOP-REAL n) st P is compact & Q is compact & HausDist (P,Q) = 0 holds
P = Q

proof end;

theorem :: HAUSDORF:42

for n being Element of NAT
for P, Q, R being non empty Subset of (TOP-REAL n) st P is compact & Q is compact & R is compact holds
HausDist (P,R) <= (HausDist (P,Q)) + (HausDist (Q,R))

proof end;